On Symmetry Through-put linking 3-4-5-6-7 Dimension's ,
Octo-Symmetry for the { 4-Dimension Quasi-Octahedron } ,
and 2 Identical { 8-D Structures } oriented Symmetrically

by Frank C. Fung ( 1st published in July, 2017. )

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Section III - Recalling the 28 Symmetric Lines in a { 7-Dimension Vector Space }

Summary for the Section :

• In this Section , we shall recall our findings on :
  • the { 28 Symmetric Lines } in a { 7-Dimension Vector Space } ,

    from the author's previous paper in published March , 2016 :

    • On { N+1 Symmetric Lines } in a { N-Dimension Vector Space }

  And based on this , we shall then set up :

  • a new { set of 35 Lines } orthogonal the current { Line-Of-Concern }
    • in the { 8-Dimension Vector Space } ,

    for our further study in this Part II of the paper .

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Recalling the { 28 Symmetric Lines in 7-D } :

• Let us recall that the author has previously published the paper :
  • On { N+1 Symmetric Lines } in a { N-Dimension Vector Space }

    in March , 2016 .

  Let us now recount some of the findings from that paper for our use in this paper .

• Let us now do a step-by-step recount on establishment of the [ 28 Symmetric Lines ] in a { 7-Dimension Vector Space } .

  STEP ONE :

  • We first set up a { 8-Dimension Vector Space } and selected this Line below :

    Line I.D.	* * *	Position Vector for 1st end-point of the Line									* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates								*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line-Of-Concern	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1

    • as our [ Line-Of-Concern ] for the time-being , so that :
      • all lines orthogonal to this [ Line-Of-Concern ] would reside in a { 7-Dimension Vector Space } .

  STEP TWO :

  • We then established that these eight (8) Lines , [ Line A-1 ] thru [ Line A-8 ] below :

    Line I.D.	* * *	Position Vector for 1st end-point of the Line									* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates								*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line-Of-Concern	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line A-1	*	Vector A-1	+1	+1	+1	+1	+1	+1	+1	-7	*	Vector A-2	-1	-1	-1	-1	-1	-1	-1	+7
    Line A-2	*	Vector A-3	+1	+1	+1	+1	+1	+1	-7	+1	*	Vector A-4	-1	-1	-1	-1	-1	-1	+7	-1
    Line A-3	*	Vector A-5	+1	+1	+1	+1	+1	-7	+1	+1	*	Vector A-6	-1	-1	-1	-1	-1	+7	-1	-1
    Line A-4	*	Vector A-7	+1	+1	+1	+1	-7	+1	+1	+1	*	Vector A-8	-1	-1	-1	-1	+7	-1	-1	-1
    Line A-5	*	Vector A-9	+1	+1	+1	-7	+1	+1	+1	+1	*	Vector A-10	-1	-1	-1	+7	-1	-1	-1	-1
    Line A-6	*	Vector A-11	+1	+1	-7	+1	+1	+1	+1	+1	*	Vector A-12	-1	-1	+7	-1	-1	-1	-1	-1
    Line A-7	*	Vector A-13	+1	-7	+1	+1	+1	+1	+1	+1	*	Vector A-14	-1	+7	-1	-1	-1	-1	-1	-1
    Line A-8	*	Vector A-15	-7	+1	+1	+1	+1	+1	+1	+1	*	Vector A-16	+7	-1	-1	-1	-1	-1	-1	-1

    • are orthogonal to the [ Line-Of-Concern ] and therefore reside in a { 7-Dimension Vector Space } ;

      and

    • would intersect one-another at an angle of [ 81.79 degrees ] ,
      • i.e. :
        • [ 81.79 degrees ] = arccos [ 1 / 7 ] .

  STEP THREE :

  • We then also established these 28 Lines , [ Line B-1 ] thru [ Line B-28 ] below :

    Line I.D.	* * *	Position Vector for 1st end-point of the Line									* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates								*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line-Of-Concern	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line B-1	*	Vector B-1	+1	+1	+1	+1	+1	+1	-3	-3	*	Vector B-2	-1	-1	-1	-1	-1	-1	+3	+3
    Line B-2	*	Vector B-3	+1	+1	+1	+1	+1	-3	+1	-3	*	Vector B-4	-1	-1	-1	-1	-1	+3	-1	+3
    Line B-3	*	Vector B-5	+1	+1	+1	+1	-3	+1	+1	-3	*	Vector B-6	-1	-1	-1	-1	+3	-1	-1	+3
    Line B-4	*	Vector B-7	+1	+1	+1	-3	+1	+1	+1	-3	*	Vector B-8	-1	-1	-1	+3	-1	-1	-1	+3
    Line B-5	*	Vector B-9	+1	+1	-3	+1	+1	+1	+1	-3	*	Vector B-10	-1	-1	+3	-1	-1	-1	-1	+3
    Line B-6	*	Vector B-11	+1	-3	+1	+1	+1	+1	+1	-3	*	Vector B-12	-1	+3	-1	-1	-1	-1	-1	+3
    Line B-7	*	Vector B-13	-3	+1	+1	+1	+1	+1	+1	-3	*	Vector B-14	+3	-1	-1	-1	-1	-1	-1	+3
    Line B-8	*	Vector B-15	+1	+1	+1	+1	+1	-3	-3	+1	*	Vector B-16	-1	-1	-1	-1	-1	+3	+3	-1
    Line B-9	*	Vector B-17	+1	+1	+1	+1	-3	+1	-3	+1	*	Vector B-18	-1	-1	-1	-1	+3	-1	+3	-1
    Line B-10	*	Vector B-19	+1	+1	+1	-3	+1	+1	-3	+1	*	Vector B-20	-1	-1	-1	+3	-1	-1	+3	-1
    Line B-11	*	Vector B-21	+1	+1	-3	+1	+1	+1	-3	+1	*	Vector B-22	-1	-1	+3	-1	-1	-1	+3	-1
    Line B-12	*	Vector B-23	+1	-3	+1	+1	+1	+1	-3	+1	*	Vector B-24	-1	+3	-1	-1	-1	-1	+3	-1
    Line B-13	*	Vector B-25	-3	+1	+1	+1	+1	+1	-3	+1	*	Vector B-26	+3	-1	-1	-1	-1	-1	+3	-1
    Line B-14	*	Vector B-27	+1	+1	+1	+1	-3	-3	+1	+1	*	Vector B-28	-1	-1	-1	-1	+3	+3	-1	-1
    Line B-15	*	Vector B-29	+1	+1	+1	-3	-3	+1	+1	+1	*	Vector B-30	-1	-1	-1	+3	+3	-1	-1	-1
    Line B-16	*	Vector B-31	+1	+1	-3	-3	+1	+1	+1	+1	*	Vector B-32	-1	-1	+3	+3	-1	-1	-1	-1
    Line B-17	*	Vector B-33	+1	-3	-3	+1	+1	+1	+1	+1	*	Vector B-34	-1	+3	+3	-1	-1	-1	-1	-1
    Line B-18	*	Vector B-35	-3	-3	+1	+1	+1	+1	+1	+1	*	Vector B-36	+3	+3	-1	-1	-1	-1	-1	-1
    Line B-19	*	Vector B-37	+1	+1	+1	-3	+1	-3	+1	+1	*	Vector B-38	-1	-1	-1	+3	-1	+3	-1	-1
    Line B-20	*	Vector B-39	+1	+1	-3	+1	-3	+1	+1	+1	*	Vector B-40	-1	-1	+3	-1	+3	-1	-1	-1
    Line B-21	*	Vector B-41	+1	-3	+1	-3	+1	+1	+1	+1	*	Vector B-42	-1	+3	-1	+3	-1	-1	-1	-1
    Line B-22	*	Vector B-43	-3	+1	-3	+1	+1	+1	+1	+1	*	Vector B-44	+3	-1	+3	-1	-1	-1	-1	-1
    Line B-23	*	Vector B-45	+1	+1	-3	+1	+1	-3	+1	+1	*	Vector B-46	-1	-1	+3	-1	-1	+3	-1	-1
    Line B-24	*	Vector B-47	+1	-3	+1	+1	-3	+1	+1	+1	*	Vector B-48	-1	+3	-1	-1	+3	-1	-1	-1
    Line B-25	*	Vector B-49	-3	+1	+1	-3	+1	+1	+1	+1	*	Vector B-50	+3	-1	-1	+3	-1	-1	-1	-1
    Line B-26	*	Vector B-51	+1	-3	+1	+1	+1	-3	+1	+1	*	Vector B-52	-1	+3	-1	-1	-1	+3	-1	-1
    Line B-27	*	Vector B-53	-3	+1	+1	+1	-3	+1	+1	+1	*	Vector B-54	+3	-1	-1	-1	+3	-1	-1	-1
    Line B-28	*	Vector B-55	-3	+1	+1	+1	+1	-3	+1	+1	*	Vector B-56	+3	-1	-1	-1	-1	+3	-1	-1

    • are orthogonal to the [ Line-Of-Concern ] and therefore reside in a { 7-Dimension Vector Space } ;

      and

    • would intersect one-another at an angle of [ 70.53 degrees ] ,
      • i.e. :
        • [ 70.53 degrees ] = arccos [ 1 / 3 ] .

  And these [ 28 Lines ] are then our { 28 Symmetric Lines in a 7-Dimension Vector Space } .

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New Findings based on the above :

• Let us now come to the recognition that :
  • the [ line-direction ] for the above 28 Lines can be re-expressed in this alternate format :
    • as per [ Line BB-1 ] in the table below .

    Line I.D.	* * *	Position Vector for 1st end-point of the Line									* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates								*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line-Of-Concern	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line B-1	*	Vector B-1	+1	+1	+1	+1	+1	+1	-3	-3	*	Vector B-2	-1	-1	-1	-1	-1	-1	+3	+3
    Line BB-1	*	Vector BB-1	+2	+2	+2	+2	+2	+2	-6	-6	*	Vector BB-2	-2	-2	-2	-2	-2	-2	+6	+6

  And that is to say :

  • the line-direction for both [ Line B-1 ] and [ Line BB-1 ] are the same in the { 8-Dimension Vector Space } .

• This then permitted us to formulate 2 additional new { Types of Lines } :
  • that are orthogonal to the original [ Line-Of-Concern ] ,

    namely :

    • the 2 types as per [ Line CC-1 ] and [ Line DD-1 ] in the table below .

    Line I.D.	* * *	Position Vector for 1st end-point of the Line									* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates								*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line-Of-Concern	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line A-1	*	Vector A-1	+1	+1	+1	+1	+1	+1	+1	-7	*	Vector A-2	-1	-1	-1	-1	-1	-1	-1	+7
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line BB-1	*	Vector BB-1	+2	+2	+2	+2	+2	+2	-6	-6	*	Vector BB-2	-2	-2	-2	-2	-2	-2	+6	+6
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line CC-1	*	Vector CC-1	+3	+3	+3	+3	+3	-5	-5	-5	*	Vector CC-2	-3	-3	-3	-3	-3	+5	+5	+5
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line DD-1	*	Vector DD-1	+4	+4	+4	+4	-4	-4	-4	-4	*	Vector DD-2	-4	-4	-4	-4	+4	+4	+4	+4

  And :

  • For the 1st new type { TYPE-CC Lines } :
    • the Position Vector for the { 1st end-point of the Line } is characterised by :
      • 5 [ +3 ]'s and 3 [ -5 ]'s in its [ 8 co-ordinates ] ;

      and

    • the Position Vector for the { 2nd end-point of the Line } is characterised by :
      • 5 [ -3 ]'s and 3 [ +5 ]'s in its [ 8 co-ordinates ] .

  • For the 2nd new type { TYPE-DD Lines } :
    • the Position Vector for the { 1st end-point of the Line } is characterised by :
      • 4 [ +4 ]'s and 4 [ -4 ]'s in its [ 8 co-ordinates ] ;

      and

    • the Position Vector for the { 2nd end-point of the Line } is characterised by :
      • 4 [ -4 ]'s and 4 [ +4 ]'s in its [ 8 co-ordinates ] .

  And the line-count for the 2 new { Types of Lines } are then as follows :

  • the [ total number of { Type-CC Lines } ] is then [ 56 ] ,
    i.e. :
    • [ 56 ] = [ 8 x 7 x 6 ] / [ 3 x 2 x 1 ]

  • the [ total number of { Type-DD Lines } ] is then [ 35 ] ,
    • i.e. :
      • [ 35 ] = one-half of { [ 8 x 7 x 6 x 5 ] / [ 4 x 3 x 2 x 1 ] }

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A Special Note here in passing :

• And we take note here in passing that :
  • The above procedure of :
    • selecting a [ Line-Of-Concern ] in a [ N-Dimension Vector Space ]

      and

    • generating different sets of { Symmetric Lines } orthogonal to the [ Line-Of-Concern ]

    is valid throughout for all values of [ N ] greater than [ 2 ] .

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Coverting the [ Type-DD Lines ] to the [ Type-D Lines ] format :

• Let us now come to the recognition that :
  • the [ line-direction ] for the above 35 [ Type-DD Lines ] can be re-expressed in this alternate format :
    • as per [ Line D-1 ] in the table below .

    Line I.D.	* * *	Position Vector for 1st end-point of the Line									* * *	Position Vector for 2nd end-point of the Line
    	*	Vector I.D.	Co-ordinates								*	Vector I.D.	Co-ordinates
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line-Of-Concern	*	Vector Z-1	+1	+1	+1	+1	+1	+1	+1	+1	*	Vector Z-2	-1	-1	-1	-1	-1	-1	-1	-1
    *********************	*	***********	***	***	***	***	***	***	***	***	*	***********	***	***	***	***	***	***	***	***
    Line DD-1	*	Vector DD-1	+4	+4	+4	+4	-4	-4	-4	-4	*	Vector DD-2	-4	-4	-4	-4	+4	+4	+4	+4
    Line D-1	*	Vector D-1	+1	+1	+1	+1	-1	-1	-1	-1	*	Vector D-2	-1	-1	-1	-1	+1	+1	+1	+1

  And that is to say :

  • the line-direction for both [ Line DD-1 ] and [ Line D-1 ] are the same in the { 8-Dimension Vector Space } .

• Thus , we can now create a new set of { 35 [ Type-D Lines ] } orthogonal to the original [ Line-Of-Concern ] ,
  • with each [ Type-D Line ] being characterised by :
    • hosting 4 [ +1 ]'s and 4 [ -1 ]'s in its [ 8 co-ordinates ]

      for both the { Position Vector for the 1st end-point of the Line } and the { Position Vector for the 2nd end-point of the Line } .

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What's Next :

• Let us now take a quick look at :
  • a full listing of the { 35 [ Type-D Lines ] } , next .

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go to the next Section : Section IV - A Listing of 35 Lines orthogonal to the [ Line-Of-Concern ]

go to the last Section : Section II - Defining the { 8-Dimension Quasi-Cube }

return to the Symmetry Through-put HomePage

Original dated 2017-7-23